Phase Response — Second-Order System

The phase-plane representation is a plot of dd/dt vs. 6 as families of curves for a given system with initial conditions dd/dt(0) and 0(0) as parameters. This plot or phase portrait, provides a useful indication of the transient response of a nonlinear system. It cannot be applied to sinusoidal or other continuing forcing functions. Furthermore, the method is limited to second-order systems or systems that can be handled as second-order systems. 

This figure shows phase lag vs. frequency ratio for a second-order system (RCL) in response to a sine-wave input (semilog coordinates). Damping ratios of 0.1 to 20.0 are given. 

The first term on the right-hand side of eqn. (11) decays away and, after a time approximately equal to 5t, the second term alone will remain. Note that this is a sine wave of the same frequency as the forcing function, but that its amplitude is reduced and its phase is shifted. This second term is called the frequency response of the system such responses are often characterised by observing how the amplitude ratio and phase lag between the input and output sine waves vary as a function of the input frequency, k. To recover the system RTD from frequency response data is more complex tnan with step or impulse tests, but nonetheless is possible. Gibilaro et al. [22] have described a short-cut route which enables low-order system moments to be determined from frequency response tests, these in turn approximately defining the system transfer function G(s) [see eqn. (A.5), Appendix 1]. From G(s), the RTD can be determined as in eqn. (8). 

The interference between different vibrations (including those of different molecules) resulting from the coherent nature of the experiment makes the analysis of VSFS spectra considerably more complicated than that of spectra recorded with linear spectroscopic techniques. However, this complexity can be exploited to provide orientational information if a complete analysis of the VSF spectrum is employed taking into account the phase relationships of the contributing vibrational modes to the sum-frequency response [15,16]. In the analysis it is possible to constrain the average orientation of the molecules at the surface by relating the macroscopic second-order susceptibility, Xs g of the system to the molecular hyperpolaiizabilities, of the individual... 

In the above conservations equations, expressions for the fluxes are required. This has already been accounted for in the energy and momentum balances (Eqs. 11 and 12, respectively) to provide second-order equations the reason being that they are highly coupled and remain general for many systems. However, understanding the fluxes and transport expressions for the material species including ions (Eq. 2) is critical in determining the resistances in the ionic and electronic phases and the overall response of the porous electrode thus, they are discussed in more detail. 

One core chiral system that shows dramatic amplification of its chiral structure is the substituted helicenes of Katz and coworkers [83]. In essence, this research cuts the helix into a number of six-helicene subunits that self-assemble (Figure 10). Only when these subunits, which look like lock washers, are prepared in optically pure form the material associates into supramolecular helical columns [84]. The assemblies have been synthesized with different amounts of substitution around the exterior. Depending on the helicenes substitution, the material exhibits hexagonally ordered soft-crystalline [84] or liquid-crystalline phases [85]. The liquid-crystalline versions of these molecules switch when electric fields are applied to neat and solution-phase samples and have been characterized as a dielectric response [85-87]. Upon association, these materials have enormous changes in their CD intensities and optical rotations [74]. In addition, this supramolecular chirality also significantly enhances the second-order nonlinear optical behavior of these materials in Langmuir-Blodgett films [88]. 

In DMA experiments, a sinusoidal stress is applied to the sample on increasing temperature.The response strain frequency is either in phase or out of phase with the stress depending on the viscoelastic properties of the system considered. These authors pre-cooled the sample down to -90 °C and then heated it up to 200 °C at 2 °C/min stress was applied at various frequencies (up to 20 Hz) in a three point bending test. As for the fresh sample, the trace presents a sharp drop of E and E" and a peak of tan 5. This main transition is attributable to a first order transition (ice melt) that occurred simultaneously with a change in heat capacity (a second order transition, glass transition). 

---